BEYOND THE ARITHMETIC

Transcription

1 BEYOND THE ARITHMETIC A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Antonio Montalbán August 2005

2 This document is in the public domain.

3 BEYOND THE ARITHMETIC Antonio Montalbán, Ph.D. Cornell University 2005 Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees, of course, preserving jump and join; we show that every finite partial ordering labeled with the classes in the generalized high/low hierarchy can be embedded into the Turing degrees; we show that every generalized high degree has the complementation property; and we show that if a Turing degree a is either 1-generic and 0 1, 2- generic and arithmetic, n-rea, or arithmetically generic, then the theory of the partial ordering of the Turing degrees below a is recursively isomorphic to true first order arithmetic. Second, we work with equimorphism types of linear orderings from the viewpoints of Computable Mathematics and Reverse Mathematics. (Two linear orderings are equimorphic if they can be embedded in each other.) Spector proved in 1955 that every hyperarithmetic ordinal is isomorphic to a recursive one. We extend his result and prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. From the viewpoint of Reverse Mathematics, we look at the strength of Fraïssé s conjecture. From our results, we deduce that Fraïssé s conjecture is sufficient and necessary to develop a reasonable theory of equimor-

4 phism types of linear orderings. Other topics we include in this thesis are the following: we look at structures for which Arithmetic Transfinite Recursion is the natural system to study them; we study theories of hyperarithmetic analysis and present a new natural example; we look at the complexity of the elementary equivalence problem for Boolean algebras; and we prove that there is a minimal pair of Kolmogorov-degrees.

5 BIOGRAPHICAL SKETCH I was born and raised in Montevideo, Uruguay. I started to take mathematics seriously when I join the math Olympiads team of Uruguay, in I spent three years with the math Olympiads solving fun and elementary math problems. With them I got to travel to competitions and meet many people from all around Latin America. In 1997, I entered the undergraduate program in Mathematics at the Universidad de la República in Montevideo, Uruguay. There I had Paula Severi and Walter Ferrer as my advisers. They are the ones who introduced me to the subject of logic. In 2000, I got my degree of Licenciado en Matemáticas and came to Cornell. iii

6 ACKNOWLEDGEMENTS I am very grateful to my thesis adviser Richard A. Shore for all his help during my time here at Cornell. I could not have had a better adviser. He gave me many interesting problems for me to look at, and he would listen to all my ideas and read all my drafts carefully and comment on them. He advised me not only about mathematics, but also about any other aspect of the academic life. I also want to thank my co-authors Noam Greenberg, Barbara F. Csima and Bjørn Kjos-Hanssen for letting me include the work we did together in this thesis. It has been fun working with them. I will not be able to list all the people that supported me in some way or another during the last five years. I will just list a few of them. On the one hand are the people that I met here in Ithaca, without whom I would not have enjoyed my time here as much. Among these people are all friends, my closer ones being Alison, Cristina, Everilis, Fabio, Fernando, Gabor, Johnny, Jorge, Mercedes, Pedro, Rafael and Yannet. I also want to mention Andrés, Carmen, Marta, Marcelo and Ricardo, who helped me out in my first few days in Ithaca. On the other hand are the people at home that, even though where many thousands of kilometers away, where always with me. These include my friends Alejandro and Aldo, and of course my family, my mother, my father and my sister that were always there to support me. iv

7 TABLE OF CONTENTS 1 General Introduction Turing Degree Theory Reverse Mathematics Computable Mathematics Effective Randomness Scattered linear orderings I Turing Degree structure 11 2 Embedding Jump upper semilattices into the Turing Degrees Introduction The Main construction The forcing notion Preservation of nonorder R G preserves the jump Decidability results Jump upper semilattices which support Jump Hierarchies Pseudo-well orderings with Jump Hierarchies Partial upper semilattices with level function Well quasiorderings The decomposition of J Putting the pieces together Adding 0 to the Language A negative answer A positive answer Uncountable jump upper semilattices A negative answer A positive answer There is no order in the Generalized High/Low Hierarchy Introduction GH-posets The main Lemma True Stages Tree Constructions Construction of B Construction of A Verifications v

8 4 Generalized High degrees have the complementation property (with Noam Greenberg and Richard A. Shore) Introduction Case One Construction Verifications Case Two Construction of the Tree The Construction Verifications Coding D into B Case Three A minimal degree below D Construction I Coding D Construction II Verifications II Embedding and coding below a 1-generic degree (with Noam Greenberg) Introduction Notation Genericity Slaman and Woodin Coding Coding Countable Sets Interpreting Arithmetic Lattice Embeddings II Reverse Mathematics Equivalence between Frass s conjecture and Jullien s theorem Introduction Basic Definitions Signed trees and h-indecomposable linear orderings Signed trees H-indecomposable linear orderings WQO(ST) implies ATR Finite decomposability FINDEC and WQO(ST) Minimal decomposition Fraïssé s conjecture Jullien s theorem Proof of the easy direction vi

9 6.5.2 Implications of JUL Minimal decomposition and the proof of Jullien s theorem Extendibility of h-indecomposable linear orderings Extendibility of ω and (ω 2 ) Extendibility of 1 + L Extendibility of m ω L m One step iteration The complement of a linear ordering The linearization Extendibility of η Another equivalent statement Indecomposable linear orderings and Theories of Hyperarithmetic Analysis Introduction Indecomposability Statement Game statements The Jump Iteration statement Summary or results Hyperarithmetic Theory Subsystems of second order arithmetic Linear orderings Between ACA 0 and 1 1-CA Models of INDEC The construction Pairs of computable structures The Game Statements JI does not imply G Ranked games The forcing notion The forcing relation Retaggings Ranked Structures and Arithmetic Transfinite Recursion (with Noam Greenberg) Introduction Reverse mathematics The classes The statements Reductions Results More Results Ordinals Equivalence over ACA vii

10 8.2.2 Proofs of arithmetic comprehension Well-founded trees Ranked Trees Reductions Proofs of arithmetic comprehension Superatomic Boolean algebras Definitions Ranked Boolean algebras Reductions Proofs of arithmetic comprehension Reduced p-groups Ranked p-groups Reductions Proofs of arithmetic comprehension Scattered and compact spaces Definitions Metrizable spaces Compact spaces Ranked spaces Reversals Proofs of arithmetic comprehension Up to equimorphism, hyperarithmetic is recursive III Computable Mathematics Up to equimorphism, hyperarithmetic is recursive Introduction General ideas of the Proof Signed Forests Natural sum of ordinals and ranks Signed forests and signed sequences The complements The construction Boolean Algebras, Tarski Invariants, and Index Sets (with Barbara F. Csima and Richard A. Shore) Introduction Definitions and Theorems Counting Quantifiers Dense Boolean Algebras Back-and-Forth relations The Σ n and the Π n cases (Theorem ) The Σ n Π n cases (Theorem ) viii

11 10.8 The Π ω+1 case (Theorem ) (Σ 0 ω+1, Π 0 ω+1) m (DB ω,ω,0, DB ω,0,0 ) Interval Algebras and the operation IV Miscellaneous Two results on effective randomness (with Barbara F. Csima and Bjørn Kjos-Hanssen) A minimal pair of K-degrees Introduction and Notation Construction of a minimal pair Non-Continuously Random reals Invariants for scattered linear orderings up to equimorphisms Introduction Background on linear orderings The Invariants Equimorphism invariants Ordering of invariants The class of invariants Minimal linear orderings Minimal ideals Examples of Invariants Open questions Bibliography 299 ix

12 Chapter 1 General Introduction The notion of what it means for a set, say of natural numbers, to be computable was known from even before computers existed; this is the principal notion in Computability Theory, also known as Recursion Theory. Perhaps a more important notion in Computability Theory is the one of computable from. A set A is said to be computable from (or recursive in) a set B, or Turing reducible to B, if there is a computable procedure that can tell whether an element is in A or not using B as an oracle, that is, we let the procedure use the information of which elements are in B. We can use this notion to measure the amount of information content that a set has; A has more information content than B if B can be computed from A. In this case we also say that A is more complex than B. We say that A and B are Turing equivalent if they can be computed from each other. Other notions of complexity are also relevant to Computability Theory. For example, a set is said to be arithmetic if it can be defined by a formula of first order arithmetic. The simplest set which can uniformly compute all the arithmetic set is 0 (ω). It is Turing equivalent to the set of all true sentences of first order arithmetic. Another important class of sets, greater than the class of arithmetic set and which contains 0 (ω), is the class of hyperarithmetic sets. A set is hyperarithmetic if it can be defined by a computable infinitary formula of arithmetic, or equivalently, if it is 1 1. This thesis contains all the research I have done during my time in Cornell. The main theme is computability theory. Sets which are beyond the arithmetic are considered in almost every chapter, as for example the ωth Turing Jump of, or other sufficiently complex hyperarithmetic sets. The thesis is divided in four parts. The first three parts contain my work in the areas of Turing Degree Theory, Reverse Mathematics, and Computable Mathematics. The fourth part contains what did not fit into any of the previous parts. Chapters 2, 3, 6, 7, 9, and 12 contain research that I have done myself. The other chapters (except the introductory one, of course) are joint work: Richard A. Shore is a co-author of Chapters 4 and 10; Noam Greenberg of 4, 5 and 8; Barbara F. Csima of 10, and 11; and Bjørn Kjos-Hanssen of 11. Since most of the chapters have been submitted for publication, each one is written as an individual paper. Hence, each chapter has its own introduction. The idea of this global introduction is to explain what the general ideas behind each of the three areas mentioned above are, and to summarize all the results included in this thesis. 1.1 Turing Degree Theory The Turing degree structure is a very natural object first studied by Kleene and Post in [KP54]. It is defined as follows. Consider P(N), the set of subsets of N 1

13 2 (the set of natural numbers). Given A, B P(N), let A T B if A is computable from B. The relation T is a quasi-ordering on P(N). As usual, this quasiordering induces an equivalence relation (A T B A T B & B T A) and a partial ordering on the equivalence classes. The equivalence classes are called Turing degrees. We use (D, T ) to denote this partial ordering. One of the main goals of Computability Theory is to understand the structure of (D, T ). We note that we chose to work with subsets of N because every finite object can be coded by a single number (using, for instance, the binary representation of the number). For example, strings, graphs, trees, simplicial complexes, group presentations, etc., if they are finite, they can be coded effectively by a natural number. Any other set where we can effectively code finite structures will work too. The first observation about the Turing degrees is that there is a least one, that we denote by 0. It is the Turing degree of the computable sets. The Turing degrees form an upper semilattice; that is, every pair of elements has a least upper bound. We denote the least upper bound of a and b by a b. Intuitively, a b contains all the information that a and b have. In the Turing degrees there is another naturally defined operation called the Turing jump (or just jump). The jump of a degree a, denoted a, is given by the degree of the Halting Problem relativized to some set in a. (Given A P(N), the Halting Problem relative to A, denoted by A, is the set of codes for computer programs, that, when run with oracle A, halt. Note that a computer program is a finite sequence of characters and hence can be encoded by a natural number.) It can be shown that the jump operation is strictly increasing (i.e., a(a < T a )) and monotonic ( a, b(a T b a T b )). A jump upper semilattice is an upper semilattice together with a strictly increasing, monotonic function. (Jump partial orderings are defined analogously.) So, we have that (D, T,, ) is a jump upper semilattice. The study of which structures can be embedded in (D, T ) is part of an ongoing program to understand the shape of the Turing degrees. It follows from the results of Kleene and Post [KP54] that every countable upper semilattice can be embedded into (D, T ). Since then, various other embeddablity results have been proved. Sacks proved in [Sac61] that every partial ordering of size at most ℵ 1 with the countable predecessor property can be embedded into (D, T ). (The countable predecessor property says that every element has at most countably many elements below it. Note that since a set can compute at most countably many different sets, (D, T ) has the countable predecessor property.) Abraham and Shore extended this result to upper semilattices of size ℵ 1 and of course the countable predecessor property in [AS86]. Moreover, they prove that the embedding can be constructed to be onto an initial segment of (D, T ). Hinman and Slaman proved, in [HS91], that every countable jump partial ordering is embeddable in (D, T, ). In Chapter 2 we prove the following: Theorem : Every countable jump upper semilattice can be embedded into the Turing Degrees (of course, preserving jump and join).

14 3 The proof of this theorem uses a variety of tools from Computability Theory and some ideas from [HS91]. First, via a forcing construction, we prove that every countable jump upper semilattice satisfying certain condition can be embed in (D, T,, ). This is enough to deduce that the quantifier free formulas that are true in (D, T,, ) are the exactly the ones that do not contradict the definition of jump upper semilattice. It then follows that the existential theory of (D, T,, ) is decidable. Then, using hyperarithmetic theory, Harrison linear orderings, Fraïssé limits, and well-quasiorderings, we show that every countable jump upper semilattice can be embedded into one having the condition mentioned above, and that hence can be embedded into the Turing degrees. The rest of Chapter 2 is dedicated to analyze extensions of this result. We prove that Theorem is not true about jump upper semilattices with 0, by proving that not every quantifier free 1-type of jump upper semilattice with 0 is realized in (D, T,,, 0). (Note that embedding a jump upper semilattice with 0 and one other generator can be expressed in terms of realizing quantifier free 1-types.) On the other hand, we show that every quantifier free 1-type of jump partial ordering with 0 is realized in (D, T,, 0). Moreover, we show that if every quantifier free type, p(x 1,..., x n ), of jump partial ordering with 0, which contains the formula x 1 0 (m) &... & x n 0 (m) for some m, is realized in (D, T,, 0), then every every quantifier free type of jump partial ordering with 0 is realized in (D, T,, 0). We also study the question of whether every jump upper semilattice with the countable predecessor property and size κ 2 ℵ 0 is embeddable in (D, T,, ). We show that for κ = 2 ℵ 0 the answer is no. For cardinals κ between ℵ 0 and 2 ℵ 0 we show that, if MA(κ) holds, then the answer is yes. (MA(κ), Martin s axiom for κ, is defined in ) The reason being essentially that Martin s axiom allows us to do the forcing construction we needed to get the embedding in theorem These last two results imply that whether every jump partial ordering (or jump upper semilattice) of size ℵ 1 and with the countable predecessor property is embeddable in (D, T,, ) or not is independent of ZFC. In [JP78], Jockusch and Posner defined the generalized high/low hierarchy with the intention of classifying the Turing degrees depending on how close a degree is to being computable, and on how close it is to computing the Halting Problem. This classification extended the already known classification of the 0 2 degrees (i.e., the degrees T 0 ) via the high/low hierarchy. For n 1 we say that a degree x is generalized low n, or GL n, if x (n) = (x 0 ) (n 1). We say that a degree x is a generalized high n degree, or GH n, if x (n) = (x 0 ) (n), and it is generalized intermediate, or GI, if n ( (x 0 ) (n 1) < T x (n) < T (x 0 ) (n)). (Note that (x 0 ) (n 1) is the lowest and (x 0 ) (n) is the highest x (n) could be.) From these classes, taking differences in the obvious way, we can define the proper classes GL 1, GL 2,..., GI,..., GH 2, GH 1 which partition the Turing degrees. (For instance GL 2 = GL 2 GL 1, GH 2 = GH 2 GH 1, etc..) When one first sees the definition

15 4 of this classes, one would think that generalized high degrees should be above generalized low degrees, or at least not below. This intuition is correct for the high/low hierarchy, but we show that it is as wrong at it could be in the generalized case. In Chapter 3, we prove the following. Theorem 3.2.3: Every finite partial ordering labeled with elements of the set {GL 1, GL 2,..., GI,..., GH 2, GH 1} can be embedded in the Turing degrees preserving the labels. Note that no condition at all is imposed on the labels. This result helps to understand how the degrees in the various classes of the generalized high/low hierarchy are located inside the structure of the Turing degrees. It follows from it that the existential theory of the Turing degrees, in the language with Turing reduction, 0, and unary relations for the classes in the generalized high/low hierarchy, is decidable, answering an open question posed by Lerman [Ler85]. We should mention that these result follow from the decidability of (D, T,,, 0). Lerman has a proof of this result, that he has not finish writing yet. Our method is very different than Lerman s and definitely simpler. Our construction uses two 0 -priority constructions, one in top of the other. Lerman s uses the very interesting, but rather complicated, framework of Iterated Trees of Strategies, which is general method for 0 (n) -priority constructions for arbitrary n. (See [LL96] for more information about this method.) By definition, generalized high degrees are the ones that have jumps which are as high as they could be. This is what makes them to be, in some sense, close to computing 0. One then wants to know what other properties of 0, are shared by the generalized high degrees too. For instance, it is known that: every countable partial order can be embedded below 0 (Kleene and Post [KP54]); there are minimal degrees below 0 (Sacks [Sac61]); 0 cups to every degree above it (Friedberg [Fri57]); every degree below 0 joins up to 0 (Robinson [Rob72], Posner and Robinson [PR81]). This properties also hold for generalized high degrees (Jockusch and Posner [JP78]; (Cooper [Coo73]) for H 1 and Jockusch [Joc77] for GH 1 ; Jockusch and Posner [JP78]; Posner [Pos77].) The property that we consider in Chapter 4 is complementation. A Turing degree a has the complementation property if the partial ordering of degrees below it is complemented, i.e., if for every b < T a, there exists c < T a such that b c = a and b c (the greatest lower bound of b and c) exists and is equal to 0. Robinson, Epstein and Posner [Rob72, Eps75, Pos77, Pos81] proved that 0 has the complementation property. Posner asked in [Pos81] if every generalized high degree had the complementation property. Noam Greenberg, Richard A. Shore and I answered this question affirmatively. Theorem 4.1.1: Every degree d GH 1 has the complementation property. As usual when dealing with generalized high degrees, rates of growth and domination properties play prominent roles in our constructions.

16 5 Another way of analyzing the structure of (D, T ) is by studying the complexity of its theory and the theory of its initial segments. For example, the theory of (D, T ), not only it is known to be undecidable (Lachlan [Lac68]), but also it is know that it is recursively isomorphic to true second order arithmetic (Simpson [Sim77]), or in other words, 1-1-equivalent to 0 (ω). For the case of the local theories, it was proved by Shore [Sho81] that the theory of (D( 0 ), T ) is recursively isomorphic to true first order arithmetic, where D( 0 ) is the set of degrees which are 0. Moreover, he proved that if a is high or if it bounds an r.e. degree, then the theory of (D( a), T ) interprets true first order arithmetic. Together with Noam Greenberg, we extend this result even further in Chapter 5. We show that, if g is a degree such that the 1-generic degrees are downwards dense below it, then the theory of (D( g), T ) interprets true first order arithmetic. (See Subsection for a definition of 1-generic.) In particular, this holds if g is 2-generic or 1-generic below 0. It follows that for almost every (in the sense of category) degree g, (D( g), T ) interprets true first order arithmetic. It also follows that if a is n-rea, or if it is arithmetically generic, then the theory of (D( g), T ) is recursively isomorphic to true first order arithmetic. To prove our result we show that 1-genericity is sufficient to find the parameters needed to code a set of degrees using Slaman and Woodin s method [SW86]. We also prove in Chapter 5 that any recursive lattice can be embedded below a 1- generic degree preserving top and bottom. 1.2 Reverse Mathematics The questions of which axioms are necessary to do mathematics is of great importance in the Foundations of Mathematics and is the main question behind Friedman and Simpson s program of Reverse Mathematics. To analyze this question formally it is necessary to fix a logical system. Reverse Mathematics deals with the system of second-order arithmetic. Second-order arithmetic, though much weaker than set theory, is rich enough to be able to express an important fragment of classical mathematics. This fragment includes number theory, calculus, countable algebra, real and complex analysis, differential equations, separable metric spaces and countable combinatorics among others. Almost all of mathematics that can be modeled with, or coded by, countable objects can be done in second-order arithmetic. The idea of Reverse Mathematics is as follows. We start by fixing a basic system of axioms. The most commonly used basic system is called RCA 0, which is closely related to Computable Mathematics. (RCA 0 contains the axioms of semirings, Σ 0 1- induction and the axiom scheme of 0 1-comprehension which essentially says that a set exists if it can be computed from sets that we already know exist.) Now, given a theorem of ordinary mathematics, the question we ask is what axioms do we need to add to the basic system to prove this theorem. It is often the case in

17 6 Reverse Mathematics that we can prove that a certain set of axioms is needed to prove a theorem by proving that the axioms follow from the theorem within some basic system. Because of this idea, this program is called Reverse Mathematics. Many different systems of axioms have been defined and studied. But a very interesting fact is that most of the theorems, whose proof-theoretic strength has been analyzed, have been proved equivalent over RCA 0 to one of five systems: RCA 0, WKL 0, ACA 0, ATR 0 and Π 1 1-CA 0 ordered from weakest to strongest. (See Section 6.1 for more information on Reverse Mathematics.) I started working in Reverse Mathematics trying to find the proof-theoretic strength of Jullien s Theorem which is a classification of the countable extendible linear orderings [Jul69]. This question was posed by Downey and Remmel in [DR00, Question 4.1]. A linear ordering is extendible if every countable partial ordering in which it is not embeddable it has a linearization in which it is not embeddable either. The proof theoretic strength of the extendibility of N, Z or Q, was analyzed in [DHLS03]. To know the proof-theoretic strength of Jullien s Theorem is also interesting because its proof seems to require more complex axioms than most of the theorems in classical mathematics. The answer that we found is the following. Theorem : Jullien s Theorem is equivalent to Fraïssé s conjecture over RCA 0 +Σ 1 1-IND. Fraïssé s conjecture (also known as Laver s Theorem [Lav71]) is the statement that says that the countable linear orderings form a well quasiordering with respect to embeddablity. (A well-quasi-ordering is a quasi-ordering without infinite descending sequences or infinite antichains.) Fraïssé s conjecture has interested logicians for many years also because of the difficulty of its proof in terms of reverse mathematics. While studying the proof theoretic strength of Jullien s theorem, we prove the extendibility of many linear orderings, including N 2 and Q, using just ATR 0 +Σ 1 1- IND. Moreover, for all these linear orderings, L, we prove that any partial ordering, P, in which L is not embeddable has a linearization, hyperarithmetic in P L, in which L is not embeddable either. We also prove that Fraissé s conjecture is equivalent, over RCA 0, to two other interesting statements. One that says that every scattered linear ordering is equimorphic to a finite sum of indecomposable linear orderings. (A linear ordering is scattered if Q cannot be embedded into it. A linear ordering L is indecomposable if whenever L embeds into a sum of linear ordering A + B, we have that L embeds either into A or into B. Two linear orderings are equimorphic if they can be embedded in each other.) The other statement says that the class of well founded labeled trees, with labels from {+, }, with very a natural order relation, is well quasiordered. This trees are called signed trees. (See Section ) Signed trees are very useful in this context because they are easier to deal with than linear orderings, from an effective point of view. The notion of signed trees and the

18 7 operation lin( ) (which assigns linear orderings to sign trees) are essentially new; although they have a similar flavor with the trees T (Ψ) used by Laver [Lav71, pag.104]. In the last section of Chapter 6 we look at a partition theorem about linear orderings that we believe is also equivalent to FRA. This theorem is due to Laver [Lav73] and, when restricted to countable linear orderings, says the following. For every countable linear ordering L there exists a natural number n L such that for every coloring of L with finitely many colors, there exists a subset of L which is equimorphic to L and is colored with at most n L many colors. We call this statement LAV. We show that RCA 0 +LAV implies FRA. We do not know whether the reverse implication holds or not. (See the end of Section 6.7 for a short discussion about this reversal.) The main question that is left open is what is the exact proof-theoretic strength of Fraïssé s conjecture (FRA). It is known that Laver s original proof of FRA can be carried out in Π 1 2-CA 0, and that since FRA is a true Π 1 2 statement, it cannot imply Π 1 1-CA 0. Shore [Sho93] proved that the fact that the class of well orderings is well quasiordered under embeddablity implies ATR 0 over RCA 0, getting as a corollary that FRA implies ATR 0. But we still do not know whether FRA could be proved using just ATR 0 (not even Π 1 1-CA 0 ), as has been conjectured by Peter Clote [Clo90], Stephen Simpson [Sim99, Remark X.3.31] and Alberto Marcone [Mar]. Chapter 7 is about hyperarithmetic second order arithmetic, also know as hyperarithmetic analysis. Definition A statement of hyperarithmetic analysis is a sentence of second order arithmetic S such that for every Y ω, the minimum ω-model of RCA 0 +S containing Y is HYP(Y), the ω-model consisting of the sets hyperarithmetic in Y. We provide an example of a mathematical theorem which is a statement of hyperarithmetic analysis. This statement, that we call INDEC, is due to Jullien [Jul69]. It says that every scattered indecomposable linear ordering is indecomposable either to the right or to the left. (A linear ordering L is indecomposable to the left (right) if for every non-trivial cut L = A + B, we have that L embeds in A (in B).) Theories of hyperarithmetic analysis have been studied in the seventies (see [Fri75], [Van77] and [Ste78]), but (to the author s knowledge) this one is the first natural (already published and purely mathematical) example of a statement of hyperarithmetic analysis. We also prove that, over RCA 0, INDEC is implied by 1 1-CA 0 and implies ACA 0, but of course, neither ACA 0, nor ACA + 0 imply it. INDEC is also unusual because is not equivalent to any of the five systems mentioned at the beginning of this section. Another interesting fact about INDEC is that is incomparable over ACA 0 to other natural statements of mathematics. This is probably the first example of previously published purely mathematical statements which are incomparable and are between ACA 0 and ATR 0. The statements we have in mind are the following: The existence of elementary equivalence invariants for Boolean Algebras, and

19 8 Ramsey Theorem. The former statement was studied by Shore [Sho04]. The latter statement, Ramsey s Theorem, has been extensively studied in the context of reverse mathematics (see [Sim99, III.7], [CJS01], or [Mil04, Chapter 7]). We introduce five other statements of hyperarithmetic analysis and study the relations among them. Four of them are related to finitely-terminating games. The fifth one, related to iterations of the Turing jump, is strictly weaker than all the other statements that we study in this chapter, as we prove using Steel s method of forcing with tagged trees. ATR 0 is the natural subsystem of second order arithmetic in which one can develop a decent theory of ordinals ([Sim99]). Together with Noam Greenberg, in Chapter 8, we investigate classes of structures which are in a sense the wellfounded part of a larger, simpler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian p-groups, and countable, compact topological spaces. The structures in all of these classes code ordinals as invariants. Using computable reductions between these classes, we show that Arithmetic Transfinite Recursion is the natural system for working with them: natural statements (such as comparability of structures in the class) are equivalent to ATR 0. The reductions themselves are also objects of interest. 1.3 Computable Mathematics Computable Mathematics deals with the computable aspects of mathematical theorems and objects. The question given a mathematical structure, which is the simplest way to represent it? is of great importance in this area. Part of our work in Computable Mathematics is related to this question. In [Spe55], Clifford Spector proved that every hyperarithmetic well ordering is isomorphic to a computable one. In less technical terms this says that if an ordinal has a representation of a certain complexity (hyperarithmetic, which is quite high) then it has a very simple (computable) representation. We prove a generalization of this result to all countable linear orderings: Theorem 9.1.2: Every hyperarithmetic linear ordering is equimorphic with a recursive one. Spector s theorem is a special case of Theorem because if a linear ordering is equimorphic to an ordinal, it is actually isomorphic to it. The proof of Theorem requires a deep analysis of the structure of the countable linear orderings modulo equimorphisms. This analysis is done by studying the structure of signed trees defined in Section It is often the case that proofs in Computable Mathematics (and also in Reverse Mathematics) give us a deeper understanding of objects from classical mathematics. This is definitely the case here.

20 9 On the way to Theorem we prove that a linear ordering has Hausdorff rank less than ω1 CK if and only if it is equimorphic to a recursive one. As a corollary of the proof we prove that given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most α ordered by embeddablity, is recursively presentable. Using a result of Ask and Knight [AK00], it is not hard to see that Theorem is true for the class of Boolean Algebras. With Noam Greenberg, we show in Section 8.7 that Theorem can also be extended to the class of p-groups. The other chapter in Computable Mathematics has to do with Boolean algebras. Tarski [Tar49] defined a way of assigning to each boolean algebra, B, an invariant inv(b) In, where In is a set of triples from N, such that two boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra up to isomorphism. In Chapter 10, with Barbara F. Csima and Richard A. Shore we analyze the complexity of the question Does B have invariant x?. For each x In we define a complexity class Γ x, that could be either Σ n, Π n, Σ n Π n, or Π ω+1 depending on x, and prove that the set of indices for computable boolean algebras with invariant x is complete for the class Γ x. Analogs of many of these results for computably enumerable Boolean algebras were proved in [Sel90] and [Sel91]. According to [Sel03] similar methods can be used to obtain the results for computable ones. Our methods are quite different and give new results as well. As the algebras we construct to witness hardness are all dense, we establish new analogous results for the complexity of various isomorphism problems for dense Boolean algebras. 1.4 Effective Randomness Within the area of Effective Randomness we are interested in the notion of K- reducibility. K-reducibility is defined with the intention of measuring the relative randomness of infinite binary strings. This reducibility was defined using a function, K, that assigns to each finite binary string the length of its shortest description, in a sense that we will specify (Section ). The idea being that if a string is random, there should not be any short way of describing it. Together with Barbara F. Csima, in Chapter 11, we construct a minimal pair of K-degrees, answering a question of Downey and Hirschfeldt. We do this by showing the existence of an unbounded nondecreasing function f which forces K- triviality in the sense that γ 2 ω is K-trivial if and only if for all n, K(γ n) K(n) + f(n) + O(1). Recently, Downey, J. Miller and Nies have started to study the complexity and

21 10 other aspects of these kind of functions [DYM]. Another notion that we are interested in is the one of non-continuously-random reals. This class of reals was introduced by Reimann and Slaman [RS]. In March 2005, Slaman gave a talk at the Annual ASL meeting in Stanford presenting their paper. Back then, they knew that there is a real of Turing degree 0 which is not continuously random, that there is one which is not 0 2, and that every real which is not continuously random belongs to Gödel s constructible universe L. With Bjørn Kjos-Hanssen, we show, in Section 11.2, that there are reals which are not continuously random all the way up the hypearithmetic hierarchy. Reimann and Slaman have recently proved that every real which are not continuously random is hypearithmetic. 1.5 Scattered linear orderings The last chapter is about the structure of the equimorphism types of scattered linear orderings and not really about Computability Theory. The ideas of this section are originated by the work on Chapter 9 were we prove that every hyperarithmetic linear ordering is equimorphic to a computable one. In this last chapter, Chapter 12, we define invariants for scattered linear orderings of arbitrary cardinality which classify them up to equimorphism. More precisely, we assign to each scattered linear ordering L a a finite sequence Inv(L) of finite trees labeled by ordinals and signs in {, +}. This assignment is an equimorphism invariant, that is, two scattered linear orderings A and B are equimorphic if and only if Inv(A) = Inv(B). We show that the definition of the embeddablity relation on the invariants is relatively simple, and that we can easily characterize the finite sequences of finite trees that correspond to invariants. Also, for each ordinal α, we explicitly describe the finite set of minimal scattered equimorphism types of Hausdorff rank α. We compute the invariants of each of these minimal types.

22 Part I Turing Degree structure 11

23 Chapter 2 Embedding Jump upper semilattices into the Turing Degrees. Published in the Journal of Symbolic Logic, volume 68 (3), year 2003, pages Introduction. We deal with the following kind of structures. Definition A partial jump upper semilattice (pjusl) is a structure J = J, J,, j where J, J is a partial ordering, is a partial binary operation and j are partial unary operation such, that for all x, y J, if x y is defined, it is the least upper bound of x and y, and if j(x) is defined then x < J j(x); and if j(y) is also defined and x J y, then j(x) J j(y). By partial operation we mean that it does not need to be defined everywhere. A jump upper semilattice (jusl) is a pjusl where j and are total operations. A jump partial ordering (jpo) is a pjusl where j is total but is undefined. Given pjusls, J 1 and J 2, an embedding of J 1 into J 2 is an injective map f : J 1 J 2 such that for all x, y J 1 : x J1 y if and only if f(x) J2 f(y); if j(x) is defined, then f(j(x)) = j(f(x)); and if x y is defined, then f(x y) = f(x) f(y). Observe that, D = D, T,,, the set of Turing degrees together with the Turing reduction, the join operation and the Turing Jump is a jusl. We address the question of which pjusls can be embedded into D. The first embeddablity result about D was proved by Kleene and Post in [KP54]. One of the things they proved there is that every finite upper semilattice can be embedded into D. Various others results have been proved. Sacks proved in [Sac61] that every partial ordering of size at most ℵ 1 with the c.p.p. can be embedded into D. (Recall that we say that a partial order has the c.p.p. or countable predecessor property if every element has at most countably many predecessors.) Abraham and Shore extended this result to upper semilattices in [AS86]. (They even embedded the upper semilattices as initial segments of D.) Hinman and Slaman, proved in 12

24 13 [HS91], that every countable jpo is embeddable in D. We prove here that every countable jusl is embeddable in the Turing degrees. We also construct a jpo of size continuum with the c.p.p. which cannot be embedded in D. For cardinals κ between ℵ 0 and 2 ℵ 0, we show that, if MA(κ) holds, then every jusl with the c.p.p. and size κ can be embedded in D. (MA(κ) is defined in ) These two last results imply that whether every jpo (or jusl) of size ℵ 1 is embeddable in D is independent of ZFC. These kinds of results are always related to decidability results. We know that the elementary theory of D, T is undecidable, as was shown by Lachlan in [Lac68]. However, it is still of interest to know which segments of the theory of D are decidable. For example, form the results of Kleene and Post in [KP54], we get that the -theory of D, T is decidable. Then Jockusch and Slaman, [JS93], showed that the -theory of D, T, is decidable. Their result is optimal in the sense that the -theory of the same structure is undecidable. This follows from the undecidability of the -theory of D, T, proved by Schmerl (see [Ler83, Corollary VII.4.6]). Another interesting result, proved by Jockusch and Soare is that the whole elementary theory of D, is decidable (see [Ler83, Exercise III.4.21]). Here, as a corollary of our main result, we get that the existential theory of D, T,, is decidable. This result is optimal too, since the -theory was recently proved undecidable by Shore and Slaman, in [SS]. About D, T,, we know that the -theory is decidable and that the theory is undecidable. But, we do not know much about the -theory. A sub case of this question, that remains open, is whether the existential theory of D, T,, 0 is decidable. The best approximation to this question is a result due to Lempp and Lerman [LL96]. They proved that every quantifier free formula, ϕ(x 1,..., x n ), in the language of D, T,, 0, that is consistent with the axioms of jpo with 0 (see for a definition of jpo with 0) and with the formula x 1 T 0 &... & x n T 0, is realized by a n-tuple of r.e. degrees. We call a type, p(x 1,..., x n ) of jpo with 0 archimedean if, for some m ω, it contains the formula x 1 T 0 (m) &... & x n T 0 (m). We prove that if every quantifier free (q.f.) archimedean type of jpo with 0 is realized in D, then every q.f. type of jpo with 0 is realized in D. It seems likely that the hypothesis of every q.f. archimedean type being realized in D can be proved using iterated trees of strategies, which is a method created by Lempp and Lerman (see, for example, [LL96]). Hinman and Slaman proved in [HS91] and [Hin99] that every q.f. archimedean 1-type of jpo with 0 is realized in D. (Actually they proved something equivalent to this. See the proof of Corollary for an explanation of the equivalence.) We extend their result and prove here that every q.f. 1-type of jpo with 0 is realized in D. We also show that this result cannot be extended to jusl with 0. More precisely, we prove that not every quantifier free 1-type of jusl with 0 is realized in D. This also implies that not every countable jusl with 0 can be embedded in D.

25 14 Outline. We start by proving that any countable pjusl which supports a jump hierarchy is embeddable in D. (We define jump hierarchies in ) We do this via a forcing construction that uses some ideas from the one that Hinman and Slaman used in [HS91]. We both simplify the construction in [HS91] and add new features to it. Then, in section 2.3, we show that certain simple pjusls support jump hierarchies and we deduce that the existential theory of D, T,, is decidable. In section 2.4 we prove our main result: Every countable jusl is embeddable in D. To do this we show that every countable jusl can be embedded into one that supports a jump hierarchy. Part of this proof uses Fraïssé limits which are somewhat similar to the geometric part of the forcing notion used by Hinman and Slaman in [HS91]. In the last two sections we study pjusls with 0 and uncountable pjusls. 2.2 The Main construction Definition Given a structure P = P, P,..., where P, P is a partial ordering, a Jump Hierarchy over P is a map H : P ω ω such that, for all x, y P, P T H(x); x P y H(x) T H(y); if x < P y then H(x) T H(y). When such an H exists, we say that P supports a jump hierarchy. This section is devoted to proving the following theorem. Theorem Every countable partial jump upper semilattice which supports a jump hierarchy can be embedded in D. We shall use a forcing construction (see [SWa]). We shall also use different kinds of codings. Here is a description of them. Definition For any X, Y, Z ω ω, and any n ω: 1. X codes Y (directly) in the nth column if X [n] = Y. (Where X [n] (m) = X( n, m ).) 2. X jump codes Y in the nth column if for all m, Y (m) = lim z X( n, m, z ); that is, for some function S and all m and z S(m), Y (m) = X( n, m, z ). S is called a Skolem function for the coding.

26 15 3. X codes Y lazily in the nth column if for all m and z, either X( n, m, z ) = 0 or X( n, m, z ) = Y (m) + 1, and for each m there is at least one z such that Y (m) + 1 = X( n, m, z ). 4. X and Y code Z lazily in the nth column if for all k, l and m ω, X( n, m, l ) = Y ( n, m, l ) = k 0 Z(m) = k 1, and for each m there is at least one l such that X( n, m, l ) = Y ( n, m, l ) = Z(m) + 1. Observation For X, Y and Z ω ω, If X codes Y directly or lazily in some column, then Y T X. If X jump codes Y in some column, then Y T X. If X and Y code Z lazily, then Z T X Y. Fix J = J, J,, j, a countable partial jump upper semilattice. Assume that J, the universe of J, is a recursive subset of ω. Let H : J ω ω be a jump hierarchy over J. In a first reading of this proof, the reader can assume that and j are total: there are no essential changes in the proof when we allow and j to be partial. We shall define a function R G : J ω ω via a forcing construction. The map x degree(r G (x)): J D is going to be the desired embedding. For each x J, R G (x) consists of: A direct code of H(x) in the 0th column. A jump coding of R G (j(x)) in the 2nd column if j(x). This jump coding has Sk G (x) as a Skolem function. A lazy coding of R G (y) in the (3y)th column for all y < J x. A lazy code of the Skolem function Sk G (y) in the (3y + 1)st column for all y such that j(y) = x. In the (3 x, z + 2)nd column, R G (x) and R G (z) code R G (x z) lazily for each z J x such that x y is defined, where x, z = min( x, z, z, x ) (it is a code for the unordered pair {x, z}), and x J z stands for x J z & z J x The forcing notion. Now we define a partial ordering P. Then we consider a generic filter G over P, and from it define R G : J ω ω.